CHAPTER 23 / Model-based experiments

State feedback by pole placement

A position-sensed mass–spring–damper with designed state feedback and a Luenberger observer. Compare true and estimated states, noise and actuator limits.

01 / THE INTUITION

First, picture it

Does doubling the desired natural frequency improve speed without any cost?

02 / THE IDEA

What the model says

The lab's mass has position x and velocity v. Its open-loop equation is x″ + 0.8x′ + 4x = u. Position and velocity form the state vector; the input is force. Try a feedback law u = −k₁x − k₂v + Nr. Substitution gives x″ + (0.8+k₂)x′ + (4+k₁)x = Nr. The gains change the coefficients of the differential equation, so they change its poles.

RELATIONx′ = Ax + Bu; ureq = −Kx̂ + Nr; x̂′ = Ax̂ + Buapplied + L(y − Cx̂)

m = 1 kg, k = 4 N/m, c = 0.8 N·s/m; x = [position, velocity], y = position + noise. Continuous controller and observer, RK4 at 1 ms, recorded every 20 ms. Observer damping is 0.8; its natural frequency is the chosen ratio times controller ωn. Uniform seeded noise is held for 20 ms. Saturation is included; delays, model mismatch and process noise are not. This is not a sampled digital implementation or a Kalman filter.

01 / STEP BY STEP

From a desired response to a polynomial

Choose a natural frequency ωn and damping ratio ζ, giving the desired denominator s² + 2ζωn s + ωn². Matching coefficients yields k₁ = ωn²−4 and k₂ = 2ζωn−0.8. The lab computes these gains rather than asking you to guess them. For ωn = 2 and ζ = 0.7, K = [0, 2]. The resulting poles are approximately −1.4 ± j1.43. Doubling ωn at fixed ζ moves both poles proportionally farther from the origin in this nominal model.

02 / STEP BY STEP

Reference scaling is not integral action

At a constant target with zero velocity, the nominal equilibrium satisfies (4+k₁)x = Nr. Choosing N = 4+k₁ = ωn² makes x = r in the unsaturated exact model. This prefilter is a model-based reference gain. It does not measure accumulated error and does not automatically reject an unknown constant load or a wrong spring coefficient. Integral augmentation would be a separate design.

The controllability test answers whether a chosen input can move every mode. It does not promise small gains or acceptable actuator effort. This two-state model is controllable, but a finite force limit can invalidate the nominal pole prediction. Inspect requested and applied force when increasing bandwidth. Pole placement chooses locations; LQR instead computes a gain from a quadratic state/input cost. Neither method removes the need to check constraints.

03 / IN PRACTICE

Make it concrete

Pin ωn = 2. Change only ωn to 4, then 6 rad/s. Compare position and requested/applied force. Keep damping and the initial state fixed.

Open the connected experiment

YOUR EXPERIMENT

Make a prediction before moving a control.

PREDICTDoes doubling the desired natural frequency improve speed without any cost?
CHANGE ONE THINGPin ωn = 2. Change only ωn to 4, then 6 rad/s. Compare position and requested/applied force. Keep damping and the initial state fixed.
OBSERVEPin the baseline. Compare the same time and inspect the physical input as well as the output.
EXPLAINK is calculated from the desired second-order polynomial. Faster nominal poles increase the required feedback and reference gains. The ±10 N actuator can clip the request, so the actual response need not follow those poles. Pole placement describes the unsaturated, exact-model linear loop.

BE CAREFUL

K is calculated from the desired second-order polynomial. Faster nominal poles increase the required feedback and reference gains. The ±10 N actuator can clip the request, so the actual response need not follow those poles. Pole placement describes the unsaturated, exact-model linear loop.

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